There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} (p^{n}-p^{k}) = |GL(n,p)| = |Aut(E)|$, where $E$ is an elementary abelian group of order $p^{n}$.
The proof I know is by proving the $p$-part and the $p^{'}$-part of the divisibility separately.
The $p^{'}$-part of the divisibility boils down to the integrality of the binomial coefficient analogues which count decompositions of a vector space over $\mathbb{Z}/(p)$ into two subspaces whose sum is the space and whose intersection is $0$.
The $p$-part of the divisibility is the divisibility statement for the order of a Sylow $p$-subgroup of $Aut(G)$, and it uses induction and the fact that a $p$-group will have fixed points whenever it acts on a set whose cardinality is a nonmultiple of $p$.
Is there a book that covers this theorem? If so, how far (and where) does the book run with it?
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2$\begingroup$ Peter Neumann's paper "Proof of a conjecture by Garrett Birkhoff and Philip Hall on the automorphisms of a finite group." blms.oxfordjournals.org/content/27/3/222 runs with this in the sense of generalising it to arbitrary finite groups. $\endgroup$– M TCommented Jun 18, 2011 at 13:52
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$\begingroup$ Thanks for the comment, but I actually meant 'running with it' in the opposite sense: not what theorems have it as a consequence, but what consequences come from it. An easy consequence, which I had seen mentioned before on MO: if $|G| = p^{n}$ and $\frac{m^{2}-m}{2} < n$, then $G$ has a normal abelian subgroup of order $p^{m}$. AKA Burnside's second $p^{a}q^{b}$ Theorem: If $|G| = p^{a}q^{b}$ and $p^{a} > q^{b}$, then $\mathrm{O}_{p}(G)$ is nontrivial, unless an exceptional case happens: (i) $p=2$ and $q$ is a Fermat prime (ii) $p=2$ and $q=7$ (iii) $p$ is a Mersenne prime and $q=2$ $\endgroup$– DavidLHardenCommented Jun 18, 2011 at 21:31
2 Answers
This is just Burnside's basis theorem. See for instance theorem 12.2.2 on page 178 of M. Hall, Jr.'s textbook on the Theory of Groups. The original reference for the phrasing in terms of automorphisms is from P. Hall (1933).
As far as where to go from it, this is roughly how automorphism groups of p-groups are calcuated in O'Brien (1992), Eick–Leedham-Green–O'Brien (2002) and the AutPGrp package of GAP. This is often useful in understand fusion systems, where the p-core of automorphism groups is under good control, and so the GL(n, p) part is the primary interest.
Another application (known to the OP, but interesting enough to describe clearly) is a result of Burnside (1905) classifying for which powers b = b(p, a, q) there is a group of order paqb with no non-identity normal p-subgroup: the classification is based simply on the orders of the automorphism groups of the p-subgroups. Burnside had an error in his analysis of the associated arithmetical condition that was corrected in Coates–Dwan–Rose (1976). Burnside's result was generalized in Glauberman (1975) and Bialostocki (1975, 1987). Many of these and further results are based on analyzing nilpotent p′-subgroups of GL(n, p), resting ultimately on the fact that that every p′-subgroup of the automorphism group of a p-group of rank n is isomorphic (including in some sense, its action) to a subgroup of GL(n, p).
Burnside, W. On groups of order pαqβ Lond. M. S. Proc. (2) 1, 388-392 (1904). JFM35.0162.01 DOI:10.1112/plms/s2-1.1.388
Burnside, W. On groups of order pαqβ (second paper). Lond. M. S. Proc. (ser 2) 2, (1905) 432-437. JFM36.0198.02 DOI:10.1112/plms/s2-2.1.432
Hall, P. A contribution to the theory of groups of prime-power order. Proc. Lond. Math. Soc., Ser. 2, 36, (1933) 29-95. Zbl0007.29102 DOI:10.1112/plms/s2-36.1.29
Glauberman, G. On Burnside's other paqb theorem. Pacific J. Math. 56 (1975), no. 2, 469–476. MR412269 URL:euclid.pjm/1102906371
Bialostocki, Arie. On products of two nilpotent subgroups of a finite group. Israel J. Math. 20 (1975), no. 2, 178–188. MR407148 DOI:10.1007/BF02757885
Coates, Martin; Dwan, Michael; Rose, John S. A note on Burnside's other pαqβ theorem. J. London Math. Soc. (2) 14 (1976), no. 1, 160–166. MR419594 DOI:10.1112/jlms/s2-14.1.160
Bialostocki, Arie. On the other pαqβ theorem of Burnside. Groups–St. Andrews 1985. Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 1, 41–49. MR879428 DOI:10.1017/S0013091500017946
O'Brien, E. A. Computing automorphism groups of p-groups. Computational algebra and number theory (Sydney, 1992), 83–90, Math. Appl., 325, Kluwer Acad. Publ., Dordrecht, 1995. MR1344923
Eick, Bettina; Leedham-Green, C. R.; O'Brien, E. A. Constructing automorphism groups of p-groups. Comm. Algebra 30 (2002), no. 5, 2271–2295. MR1904637 DOI10.1081/AGB-120003468
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2$\begingroup$ Thanks to Vipul for his nice page on the Burnside result. groupprops.subwiki.org/wiki/… $\endgroup$ Commented Jun 19, 2011 at 1:49
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$\begingroup$ I could be wrong but doesn't Burnside's Basis Theorem only deal with the $p'$-automorphisms? (I think David hints at this in his question.) Also, I am slightly unsure that the result which David states is true! Are we sure that the given bound holds for $|Aut P|$ as opposed to $|Out P|$ (see the paper by Neumann "Proof of a conjecture by Garrett Birkhoff...")? $\endgroup$ Commented Nov 23, 2012 at 10:43
Berkovich's Groups of Prime Power Order, Volume 1 would be a good reference, especially chapter $6$. Theorem $6.9$ generalizes what you mentioned above by calculating directly $|\mathrm{Aut}(G)|$.
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3$\begingroup$ Theorem 6.9 is for abelian groups. $\endgroup$ Commented Jun 18, 2011 at 22:48